THE DOUBTER product of (rational) primes in only one way. From this theorem springs all the intricate theory of divisibility for rational integers. Unfortunately the fundamental theorem does not hold in all algebraic number fields of degree greater than one, and the result is chaos. To give an instance (it is the stock example usually exhibited in text-books on the subject), in the field F[ V — 5] we have 6 = 2X3 = (1 + V~5) X (1 - V~5); each of 2, 3,1 -f V - 5,1 - V~^5 is a prime in this field (as may be verified with some ingenuity), so that 6, in this field, is not uniquely decomposable into a product of primes. It may be stated here that Kronecker overcame this difficulty by a beautiful method which is too detailed to be explained untechnically, and that Dedekind did likewise by a totally different method which is much easier to grasp, and which will be noted when we consider his life. Dedekind's method is the one in widest use to-day, but this does not imply that Kro- necker's is less powerful, nor that it will not come into favour when more arithmeticians become familiar with it. In his dissertation of 1845 Kronecker attacked the theory of the units in certain special fields - those defined by the equa- tions arising from the algebraic formulation of Gauss' problem to divide the circumference of a circle into n equal parts or, what is the same, to construct a regular polygon of n sides, We can now close up one part of the account opened by Fennat. In struggling to prove Fennat's "Last Theorem* that x" -f yn = zn is impossible in rational integers x, y, z (none zero) if n is an integer greater than 2, arithmeticians took what looks like a natural step and resolved the left-hand side, a?n -f- yn, into its n factors of the first degree (as is done in the usual second course of school algebra). This led to the exhaustive investigation of the algebraic number field mentioned above in connexion with Gauss' problem - after serious but readily understandable mistakes had been made. The problem at first was studded with pitfalls, into which many a competent mathematician and at least one great one - Cauchy - tumbled headlong. Cauchy assumed as a matter of H 2 521